The world is everything that is the case.
A fact is not merely a thing that exists, but a configuration of things standing in relation to one another. The logical structure of the world is not found in its objects but in the Sachverhalte — the states of affairs that obtain. To understand the world logically is to understand how these states compose into the totality we call reality.
Consider: the proposition p ∧ q does not describe two isolated truths but their conjunction — the fact that they hold simultaneously. The world is built from such conjunctions, an immense lattice of interlocking facts whose structure is the proper subject of logic.
Completeness is the first demand of logic. It is not sufficient that certain facts obtain; we must know that these are all the facts. The totality of facts determines what is the case, and equally, what is not the case. For every proposition p, either p or ¬p obtains. There is no gap, no silence in the logical fabric of the world.
This principle of bivalence — that every meaningful proposition must be determinately true or false — is the bedrock upon which the entire edifice of classical logic rests. It is not a discovery about the world but a condition for the possibility of saying anything about it at all.
A state of affairs (Sachverhalt) is a combination of objects. The existence of such a combination is a fact. If we write ∃x(Fx ∧ Gx), we assert that there exists at least one object that participates simultaneously in two states of affairs — that bears both properties F and G.
The logical notation does not merely describe reality; it mirrors its structure. The connectives ∧, ∨, → are not additions to the world but reflections of how facts relate within it. In this sense, logic is the scaffolding of existence — invisible, necessary, and beautiful in its austerity.
A picture is a model of reality. In the picture and the pictured, there must be something identical in order that the one can be a picture of the other at all. What a picture must have in common with reality in order to depict it is its logical form — the form of reality.
This is the profound insight: a proposition like ∀x(Px → Qx) is not a description of reality in the way a photograph is. It is a logical picture — sharing not colors or shapes with reality, but structure. The arrow → in our formula mirrors the real relation between P-things and Q-things. Language reaches out to touch the world through form alone.
The limits of my language mean the limits of my world. Logic pervades the world; the boundaries of the world are also its boundaries. We cannot say in logic: “The world has this in it, and this, but not that.” For that would apparently presuppose that we exclude certain possibilities, and this cannot be the case, since otherwise logic would have to get outside the limits of the world.
What can be shown cannot be said. The propositions of logic are tautologies — they say nothing about the world, yet they show the logical form that the world and language share. In the end, the deepest truths are not spoken but displayed, manifest in the structure of our discourse like the grain in marble — always present, never stated, silently sustaining everything built upon it.